Properties

Label 2-189618-1.1-c1-0-3
Degree $2$
Conductor $189618$
Sign $1$
Analytic cond. $1514.10$
Root an. cond. $38.9115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s − 4·7-s + 8-s + 9-s − 2·10-s − 11-s − 12-s − 4·14-s + 2·15-s + 16-s − 17-s + 18-s − 4·19-s − 2·20-s + 4·21-s − 22-s − 24-s − 25-s − 27-s − 4·28-s + 2·29-s + 2·30-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.872·21-s − 0.213·22-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.365·30-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189618\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1514.10\)
Root analytic conductor: \(38.9115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.118437415\)
\(L(\frac12)\) \(\approx\) \(1.118437415\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.01405252693218, −12.65693602873522, −12.30160726573475, −11.76680861398480, −11.31046708991283, −10.90382087544122, −10.38041514335458, −9.930766184958047, −9.394408640320890, −8.923353272346501, −8.147962409493779, −7.729080002438609, −7.279085741841911, −6.622322030639193, −6.222149583794654, −6.042542487843016, −5.243023078618889, −4.633315884012958, −4.220161569270971, −3.706725739969240, −3.219854354717375, −2.597578542201910, −2.058432903637685, −0.9659904988375512, −0.3229225669928622, 0.3229225669928622, 0.9659904988375512, 2.058432903637685, 2.597578542201910, 3.219854354717375, 3.706725739969240, 4.220161569270971, 4.633315884012958, 5.243023078618889, 6.042542487843016, 6.222149583794654, 6.622322030639193, 7.279085741841911, 7.729080002438609, 8.147962409493779, 8.923353272346501, 9.394408640320890, 9.930766184958047, 10.38041514335458, 10.90382087544122, 11.31046708991283, 11.76680861398480, 12.30160726573475, 12.65693602873522, 13.01405252693218

Graph of the $Z$-function along the critical line